Gauss’ Law áElectric Field Lines / Electric Field Vectors áElectric Flux áGauss’ Law áUse of Gauss’ Law and Gaussian Surfaces áElectrostatic Equilibrium #Conductors #Non Conductors

Electric Field Vectors and Lines

Electric Force and Acceleration áThe electric force is á given by  F = qE áThe acceleration by a  q m E

áA measure of the amount of electric field through an area perpendicular to the field áThe “number” of field lines through the area. Electric Flux

Definition

Flux Picture

Area Vector Define Area Vector

Definition of symbols A = Area (always positive number) n = Unit vector. Its direction corresponds to the orientation of the area Forms a right handed system

Dot product Definition of Flux áElectric Flux áNumber of Field lines áthrough Perpendicular surface

Flux through closed surface áFlux through a closed á closed ásurface from an á external source is zero

Closed Surface Picture

Surface Area Element

Flux through Curved Surface  E  d A surface  E  d A  EdA Cos    A  dA surface 

Spherical Surface

Gaussian Surface áGaussian Surface defined as áSurface # surrounding # surrounding charge # magnitudeconstant # where magnitude of Electric Field is constant or zero direction Area vectors # the direction of Electric Field is same as the Area vectors of the surface symmetry # thus same symmetry as charge distribution

Flux through any closed surface surrounding a charge is the same

Gauss' Law I  E  d A Gaussian surface   Er   dA Gaussian surface   Er   dA Gaussian surface   Er   4  r 2

Gauss' Law III  k Q r 2 4  r 2  4  kQ  Q  0 Using Coulombs Law for a point charge

Gauss' Law II Gauss’ Law  E  d A Gaussian surface   Q  0

To Find Electric Field of Given Charge Distribution Surface + Charge Field Use of Gauss' Law

Closed Surfaces

Coulombs Law from Gauss' Law I Gauss' Law Coulombs' Law

Coulombs Law from Gauss' Law I

Electrostatic Equilibrium for objects in an external Electric Field áConductors # No net motion of charge within conductor áNon Conductors # in non conductors there is no movement of charge # therefore always have equilibrium

At Electrostatic Equilibrium á Electric Field is zero within conductor á Any excess charge on an isolated conductor must be on its surface # accumulates at points where radius of curvature is greatest

# is perpendicular to conductors surface # has magnitude = qsurface density / permitivity Electric Field just outside conductor

Electric Field inside conductor á Net Electric Field is zero inside, á otherwise Net Electric Force on charges á which then accelerate and move charges (on the average)

Why is the Charge on the Surface? Q E=0 Gaussian Surface 1 Gaussian Surface 2 Use Gauss’ Theorem Why is the charge on the surface?

Answer Charge must be between surface 1 and surface 2 (why?) Therefore must be on the surface of object

What is Electric Field on surface?

1 2 3 Zero Flux through 2 Zero Flux through 3 Only Flux through 1 E Answer

Answer 2 Q inside cylinder  0  E  d A   Er   dA disk 1   Er   A  Er    Q inside cylinder A  0   r    0

Answer 3 Direction of Field? áMust be orthogonal to surface áotherwise there will be net motion on surface

magnitude of electric field distance from center of charged conductor radius of conductor Graph of Field v. Position

á In external field conductor polarized á becomes polarized  Induced  Induced Electric Field from the surface must cancel external Electric Field inside conductor Conductor in Electric Field

Induced Field E E E E qq qq qq qq qq qq E induced

áIf the conductor has a net charge áthen it is also a source of an Electric Field áthat combines with the external field áproducing a resultant field áexternal to the conductor Charged Conductor

Electric Field inside Cavities Cavities of Conductors Electric Fields inside Cavities of Conductors Gaussian Surface Cavity

Analysis 1 áTotal charge within Gaussian surface must be zero áOtherwise there is an Electric Field inside the conductor around the cavity

á Therefore NO charge on surface of cavity á Can enlarge cavity so that conductor is hollow á Faraday cage Analysis 2

Radio reception over some bridges Thought Question

Electric Field inside Nonconductor Electric Field inside non conductor?

magnitude of electric field distance from center of charged non conductor radius of non conductor Graph of Field v. Position

Field Above Conductor Field above surface of charged conductor Does not depend on thickness of conductor E  Q A  0    0

Field Above Very Thin Nonconductor Field above surface of charged nonconductor